Finite Group Databasebeta

← #7623521

Subgroups — #7623521

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60673C211yesmaximal, characteristic
2C2×#3170C211yesmaximal, characteristic, complemented
3#33818C214yesmaximal, complemented
4#33797C212yesmaximal, complemented
5#33796C211yesmaximal, characteristic, complemented
6#33793C211yesmaximal, characteristic, complemented
7#33776C212yesmaximal, complemented
8#33775C211yesmaximal, characteristic, complemented
9#33772C211yesmaximal, characteristic, complemented
10#33762C214yesmaximal, complemented
11#33751C212yesmaximal, complemented
12#33749C212yesmaximal, complemented
13#33677C214yesmaximal, complemented
14#33676C214yesmaximal, complemented
15#33418C211yesmaximal, characteristic, complemented
16#3444—21no—
17C2×#533—21no—
18#3170—21no—
19#3170—21no—
20#3170—21no—
21#3170—21no—
22#3170C2212yes—
23#3170C2212yes—
24#3170C2212yes—
25#3170C2212yes—
26#3170C2211yescharacteristic
27#3170C2211yescharacteristic
28#3169C2211yescharacteristic
29#3169C2211yescharacteristic
30#3168—21no—
31#3168—21no—
32#3168C2212yes—
33#3168C2212yes—
34#3167—21no—
35#3167—21no—
36#3160C2211yescharacteristic
37C2×#500C2211yescharacteristic
38#3147C2211yescharacteristic
39C2×#494C2211yescharacteristic
40#3131—21no—
41C2×#487—21no—
42#3050—21no—
43#3050C2212yes—
44C2×#467—21no—
45C2×#467C2212yes—
46C4×#135—21no—
47C4×#135C2212yes—
48C2×C4×Q16—21no—
49C2×C4×Q16C2212yes—
50C22×C4×C8C2211yescharacteristic
51#2254C2212yescomplemented
52#2252C2212yescomplemented
53#2248C2212yescomplemented
54#2247C2212yescomplemented
55#2241C2212yescomplemented
56#2241C2212yescomplemented
57#2241C2212yescomplemented
58#2240C2212yescomplemented
59#2237C2218yes—
60#2236C2214yes—
61#2233C2214yes—
62#2229C2218yescomplemented
63#2227C2214yescomplemented
64#2226C2214yescomplemented
65#2220C2218yes—
66#2219C2218yes—
67#2211C2218yescomplemented
68#2209C2214yescomplemented
69#2208C2214yescomplemented
70#2189C2218yes—
71#2188C2214yes—
72#2185C2214yes—
73#2176C2218yescomplemented
74#2175C2218yescomplemented
75#2163C2212yes—
76#2163C2212yes—
77#2162C2211yescharacteristic
78#2162C2211yescharacteristic
79#2162C2212yes—
80#2150C2212yes—
81#2112C2212yes—
82C4×M32C2212yes—
83C2×#135—21no—
84C2×#135C2312yes—
85C22×Q16—21no—
86C22×Q16C2312yes—
87C23×C8D411yescharacteristic
88C23×C8C2311yescharacteristic
89#533—21no—
90#533—21no—
91#533—21no—
92#533—22no—
93#533—22no—
94#533—22no—
95#533—22no—
96#533—41no—
97#533—41no—
98#533—41no—
99#533—41no—
100#532—21no—
101#532—21no—
102#532—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105#529—22no—
106#529—22no—
107#529—41no—
108#529—41no—
109#529—41no—
110#529—41no—
111C2×#125—41no—
112#525—22no—
113#525—41no—
114#525—41no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—41no—
117C4×#41—22no—
118C4×#41—41no—
119C4×#41—41no—
120C2×C4×Q8—22no—
121C2×C4×Q8—41no—
122C2×#117—41no—
123C22×C42D411yescharacteristic
124C2×Q32C2318yes—
125C2×SD32C2318yes—
126C2×M32C2314yes—
127#500—21no—
128#500—21no—
129#500C2312yes—
130#500C2312yes—
131#500C2311yescharacteristic
132#500C2311yescharacteristic
133#499—21no—
134#499C2312yes—
135#498—21no—
136#495—21no—
137#495C2312yes—
138#494—21no—
139#494—21no—
140#494C2312yes—
141#494C2312yes—
142#494C2311yescharacteristic
143#493C2311yescharacteristic
144#492—21no—
145#488—22no—
146#488—22no—
147#487—22no—
148#487—22no—
149#487—22no—
150#487—22no—
151#487—21no—
152#487—21no—
153#486—21no—
154#486—21no—
155#467—21no—
156#467—21no—
157#467—22no—
158#467—22no—
159#467—22no—
160#467—22no—
161#467—22no—
162#467—22no—
163#467C2312yes—
164#467C2312yes—
165#467C2314yes—
166#467C2314yes—
167#466—21no—
168#466—21no—
169#466C2312yes—
170#466C2312yes—
171#465—22no—
172#465—22no—
173#465—22no—
174#465—22no—
175C4×Q16—21no—
176C4×Q16—21no—
177C4×Q16—22no—
178C4×Q16—22no—
179C4×Q16—22no—
180C4×Q16—22no—
181C4×Q16—22no—
182C4×Q16—22no—
183C4×Q16C2312yes—
184C4×Q16C2312yes—
185C4×Q16C2314yes—
186C4×Q16C2314yes—
187C4×SD16—22no—
188C4×SD16—22no—
189C4×SD16—22no—
190C4×SD16—22no—
191C4×D8—21no—
192C4×D8—21no—
193C4×D8C2312yes—
194C4×D8C2312yes—
195#427—21no—
196#427C2312yes—
197C2×#107—21no—
198C2×#107C2312yes—
199#416—22no—
200#416—22no—
201C2×#103—22no—
202C2×#103—22no—
203C2×C4×C8D411yescharacteristic
204C2×C4×C8D411yescharacteristic
205C2×C4×C8D411yescharacteristic
206C2×C4×C8D411yescharacteristic
207C2×C4×C8D411yescharacteristic
208C2×C4×C8D411yescharacteristic
209C2×C4×C8C2311yescharacteristic
210C2×C4×C8C2311yescharacteristic
211C2×C4×C8C2311yescharacteristic
212C2×C4×C8C2311yescharacteristic
213C2×C4×C8C2311yescharacteristic
214C2×C4×C8C2311yescharacteristic
215#367C2318yes—
216#366C2318yes—
217C4⋊C16C2312yes—
218C4⋊C16C2312yes—
219C4⋊C16C2314yes—
220#358C2318yes—
221#358C2318yes—
222#358C2318yes—
223#357C2318yes—
224C22⋊C16C2312yes—
225C22⋊C16C2312yes—
226C22⋊C16C2314yes—
227#346C2314yes—
228C4×C16C2312yes—
229C4×C16C2312yes—
230C2×#41—22no—
231C2×#41—41no—
232C2×#41—41no—
233C22×Q8—22no—
234C22×Q8—41no—
235C23×C4D811yescharacteristic
236C23×C4D811yescharacteristic
237C23×C4C2×D411yescharacteristic
238#135—44no—
239#135—44no—
240C2×Q16—22no—
241C2×Q16—22no—
242C2×Q16—22no—
243C2×Q16—21no—
244C2×Q16—44no—
245C2×Q16—44no—
246C2×Q16C2414yes—
247C2×Q16C2412yes—
248C2×SD16—22no—
249C2×SD16—22no—
250C2×D8—21no—
251C2×D8C2412yes—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8C2×D411yescharacteristic
255C22×C8C2×D411yescharacteristic
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2411yescharacteristic
264C22×C8C2411yescharacteristic
265C22×C8C2411yescharacteristic
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—21no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275C4⋊Q8—41no—
276C4⋊Q8—41no—
277#127—21no—
278#126—42no—
279#126—42no—
280#125—22no—
281#125—42no—
282#125—42no—
283#125—42no—
284#125—42no—
285#125—41no—
286#125—41no—
287#124—22no—
288#124—41no—
289#124—41no—
290#123—42no—
291#123—42no—
292#123—42no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—42no—
298C22⋊Q8—42no—
299C22⋊Q8—42no—
300C22⋊Q8—42no—
301C22⋊Q8—42no—
302C22⋊Q8—42no—
303C22⋊Q8—41no—
304C22⋊Q8—41no—
305C22⋊Q8—41no—
306C22⋊Q8—41no—
307#121—22no—
308#121—22no—
309#121—41no—
310#121—41no—
311#121—41no—
312#121—41no—
313#120—42no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—41no—
325C4×Q8—41no—
326C4×Q8—41no—
327C4×Q8—41no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4—42no—
334C4×D4—42no—
335C4×D4—42no—
336C4×D4—41no—
337C4×D4—41no—
338C4×D4—41no—
339C4×D4—41no—
340#117—22no—
341#117—42no—
342#117—42no—
343#117—42no—
344#117—42no—
345#117—41no—
346#117—41no—
347#117—42no—
348#117—42no—
349#117—42no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C22⋊C4—42no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42D811yescharacteristic
360C2×C42D811yescharacteristic
361C2×C42D811yescharacteristic
362C2×C42D811yescharacteristic
363C2×C42D811yescharacteristic
364C2×C42D811yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371Q32—48no—
372SD32—48no—
373M32—44no—
374C2×C16C2414yes—
375C2×C16C2414yes—
376#107—22no—
377#107—22no—
378#107—22no—
379#107—21no—
380#107—21no—
381#107C2414yes—
382#107C2412yes—
383#107C2412yes—
384#106—22no—
385#106—22no—
386#103—24no—
387#103—24no—
388#103—24no—
389#103—24no—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—22no—
394#102—22no—
395#102—22no—
396#102—22no—
397#102—22no—
398C4×C8C2×D411yescharacteristic
399C4×C8C2×D411yescharacteristic
400C4×C8C2×D411yescharacteristic
401C4×C8C2×D411yescharacteristic
402C4×C8C2×D411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2×D411yescharacteristic
410C4×C8C2411yescharacteristic
411C4×C8C2411yescharacteristic
412C4×C8C2411yescharacteristic
413C4×C8C2411yescharacteristic
414C24C2×D811yescharacteristic
415#41—44no—
416#41—44no—
417#41—84no—
418#41—84no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—41no—
424C2×Q8—41no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—84no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—41no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4C2×D811yescharacteristic
445C22×C4C2×D811yescharacteristic
446C22×C4C2×D811yescharacteristic
447C22×C4C2×D811yescharacteristic
448C22×C4C2×D811yescharacteristic
449C22×C4C2×D811yescharacteristic
450C22×C4C2×D811yescharacteristic
451C22×C4C2×D811yescharacteristic
452C22×C4C2×D811yescharacteristic
453C22×C4C2×D811yescharacteristic
454C22×C4C2×D811yescharacteristic
455C22×C4C2×D811yescharacteristic
456C22×C4C2×D811yescharacteristic
457C22×C4C2×D811yescharacteristic
458C22×C4C2×D811yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462Q16—28no—
463Q16—24no—
464Q16—44no—
465Q16—44no—
466Q16—44no—
467Q16—42no—
468SD16—44no—
469SD16—44no—
470D8—24no—
471D8—42no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8C22×D411yescharacteristic
485C2×C8C22×D411yescharacteristic
486C2×C8C22×D411yescharacteristic
487C2×C8C22×D411yescharacteristic
488C2×C8C22×D411yescharacteristic
489C2×C8C22×D411yescharacteristic
490C2×C8C22×D411yescharacteristic
491C2×C8C2511yescharacteristic
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42—44no—
532C42C2×D811yescharacteristic
533C42C2×D811yescharacteristic
534C42C2×D811yescharacteristic
535C42C2×D811yescharacteristic
536C42C2×D811yescharacteristic
537C42C2×D811yescharacteristic
538C42C2×D811yescharacteristic
539C42C2×D811yescharacteristic
540C42C2×D811yescharacteristic
541C42C2×D811yescharacteristic
542C42C2×D811yescharacteristic
543C42C2×D811yescharacteristic
544C42C22×D411yescharacteristic
545C42C22×D411yescharacteristic
546C42C22×D411yescharacteristic
547C42C22×D411yescharacteristic
548C16—24no—
549C16—24no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23C22×D811yescharacteristic
555C23C22×D811yescharacteristic
556C23C22×D811yescharacteristic
557Q8—42no—
558Q8—44no—
559Q8—42no—
560Q8—44no—
561Q8—84no—
562Q8—84no—
563Q8—82no—
564Q8—84no—
565Q8—82no—
566D4—42no—
567D4—42no—
568D4—84no—
569D4—84no—
570D4—84no—
571D4—82no—
572D4—82no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4#51014yes—
604C2×C4C22×D811yescharacteristic
605C2×C4C22×D811yescharacteristic
606C2×C4C22×D811yescharacteristic
607C2×C4C22×D811yescharacteristic
608C2×C4C22×D811yescharacteristic
609C2×C4C22×D811yescharacteristic
610C2×C4C22×D811yescharacteristic
611C2×C4C22×D811yescharacteristic
612C2×C4C22×D811yescharacteristic
613C2×C4C22×D811yescharacteristic
614C2×C4C22×D811yescharacteristic
615C2×C4C23×D411yescharacteristic
616C8—21no—
617C8—21no—
618C8—21no—
619C8—21no—
620C8—21no—
621C8—21no—
622C8#57512yes—
623C8#58512yes—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C2×#51012yes—
637C22C23×D811yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4C2×#51012yes—
652C4C2×#51012yes—
653C4#341412yes—
654C4#344911yescharacteristic
655C4C2×#57511yescharacteristic
656C2—22no—
657C2—41no—
658C2—41no—
659C2—162no—
660C2#6094011yescharacteristic
661C2C22×#51011yescharacteristic
662C2C2×#341411yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#60673C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#3170C2semidirect
3. #33818C2semidirect
4. #33797C2semidirect
5. #33796C2semidirect
6. #33793C2semidirect
7. #33776C2semidirect
8. #33775C2semidirect
9. #33772C2semidirect
10. #33762C2semidirect
11. #33751C2semidirect
12. #33749C2semidirect
13. #33677C2semidirect
14. #33676C2semidirect
15. #33418C2semidirect
51. #2254C22semidirect
52. #2252C22semidirect
53. #2248C22semidirect
54. #2247C22semidirect
55. #2241C22semidirect
56. #2241C22semidirect
57. #2241C22semidirect
58. #2240C22semidirect
62. #2229C22semidirect
63. #2227C22semidirect
64. #2226C22semidirect
67. #2211C22semidirect
68. #2209C22semidirect
69. #2208C22semidirect
73. #2176C22semidirect
74. #2175C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#606732
C2×#31702
#338182
#337972
#337962
#337932
#337762
#337752
#337722
#337622
#337512
#337492
#336772
#336762
#334182